Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2592,2,Mod(1,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,4,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.456850\) of defining polynomial
Character \(\chi\) \(=\) 2592.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.64575 q^{5} +4.37780 q^{7} -3.58258 q^{11} -6.58258 q^{13} +1.73205 q^{17} -2.55040 q^{19} +7.58258 q^{23} +2.00000 q^{25} +6.10985 q^{29} +8.75560 q^{31} +11.5826 q^{35} +2.58258 q^{37} -1.82740 q^{41} +2.55040 q^{43} +8.00000 q^{47} +12.1652 q^{49} +1.82740 q^{53} -9.47860 q^{55} +8.00000 q^{59} -1.41742 q^{61} -17.4159 q^{65} +2.55040 q^{67} +0.417424 q^{71} -6.16515 q^{73} -15.6838 q^{77} +9.47860 q^{79} +15.1652 q^{83} +4.58258 q^{85} -12.3151 q^{89} -28.8172 q^{91} -6.74773 q^{95} -5.16515 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{11} - 8 q^{13} + 12 q^{23} + 8 q^{25} + 28 q^{35} - 8 q^{37} + 32 q^{47} + 12 q^{49} + 32 q^{59} - 24 q^{61} + 20 q^{71} + 12 q^{73} + 24 q^{83} + 28 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.64575 1.18322 0.591608 − 0.806226i \(-0.298493\pi\)
0.591608 + 0.806226i \(0.298493\pi\)
\(6\) 0 0
\(7\) 4.37780 1.65465 0.827327 − 0.561721i \(-0.189860\pi\)
0.827327 + 0.561721i \(0.189860\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.58258 −1.08019 −0.540094 − 0.841605i \(-0.681611\pi\)
−0.540094 + 0.841605i \(0.681611\pi\)
\(12\) 0 0
\(13\) −6.58258 −1.82568 −0.912839 − 0.408320i \(-0.866115\pi\)
−0.912839 + 0.408320i \(0.866115\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205 0.420084 0.210042 − 0.977692i \(-0.432640\pi\)
0.210042 + 0.977692i \(0.432640\pi\)
\(18\) 0 0
\(19\) −2.55040 −0.585102 −0.292551 − 0.956250i \(-0.594504\pi\)
−0.292551 + 0.956250i \(0.594504\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.58258 1.58108 0.790538 − 0.612413i \(-0.209801\pi\)
0.790538 + 0.612413i \(0.209801\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.10985 1.13457 0.567286 − 0.823521i \(-0.307994\pi\)
0.567286 + 0.823521i \(0.307994\pi\)
\(30\) 0 0
\(31\) 8.75560 1.57255 0.786276 − 0.617875i \(-0.212006\pi\)
0.786276 + 0.617875i \(0.212006\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.5826 1.95781
\(36\) 0 0
\(37\) 2.58258 0.424573 0.212286 − 0.977207i \(-0.431909\pi\)
0.212286 + 0.977207i \(0.431909\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.82740 −0.285392 −0.142696 − 0.989767i \(-0.545577\pi\)
−0.142696 + 0.989767i \(0.545577\pi\)
\(42\) 0 0
\(43\) 2.55040 0.388933 0.194466 − 0.980909i \(-0.437703\pi\)
0.194466 + 0.980909i \(0.437703\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 12.1652 1.73788
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.82740 0.251013 0.125506 − 0.992093i \(-0.459944\pi\)
0.125506 + 0.992093i \(0.459944\pi\)
\(54\) 0 0
\(55\) −9.47860 −1.27809
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −1.41742 −0.181483 −0.0907413 − 0.995874i \(-0.528924\pi\)
−0.0907413 + 0.995874i \(0.528924\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −17.4159 −2.16017
\(66\) 0 0
\(67\) 2.55040 0.311581 0.155791 − 0.987790i \(-0.450208\pi\)
0.155791 + 0.987790i \(0.450208\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.417424 0.0495392 0.0247696 − 0.999693i \(-0.492115\pi\)
0.0247696 + 0.999693i \(0.492115\pi\)
\(72\) 0 0
\(73\) −6.16515 −0.721576 −0.360788 − 0.932648i \(-0.617492\pi\)
−0.360788 + 0.932648i \(0.617492\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.6838 −1.78734
\(78\) 0 0
\(79\) 9.47860 1.06643 0.533213 − 0.845981i \(-0.320984\pi\)
0.533213 + 0.845981i \(0.320984\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.1652 1.66459 0.832296 − 0.554332i \(-0.187026\pi\)
0.832296 + 0.554332i \(0.187026\pi\)
\(84\) 0 0
\(85\) 4.58258 0.497050
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.3151 −1.30539 −0.652697 − 0.757619i \(-0.726362\pi\)
−0.652697 + 0.757619i \(0.726362\pi\)
\(90\) 0 0
\(91\) −28.8172 −3.02086
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.74773 −0.692302
\(96\) 0 0
\(97\) −5.16515 −0.524442 −0.262221 − 0.965008i \(-0.584455\pi\)
−0.262221 + 0.965008i \(0.584455\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2592.2.a.w.1.4 yes 4
3.2 odd 2 2592.2.a.v.1.2 ✓ 4
4.3 odd 2 2592.2.a.v.1.3 yes 4
8.3 odd 2 5184.2.a.ce.1.1 4
8.5 even 2 5184.2.a.cd.1.2 4
9.2 odd 6 2592.2.i.bh.1729.3 8
9.4 even 3 2592.2.i.bg.865.1 8
9.5 odd 6 2592.2.i.bh.865.3 8
9.7 even 3 2592.2.i.bg.1729.1 8
12.11 even 2 inner 2592.2.a.w.1.1 yes 4
24.5 odd 2 5184.2.a.ce.1.4 4
24.11 even 2 5184.2.a.cd.1.3 4
36.7 odd 6 2592.2.i.bh.1729.2 8
36.11 even 6 2592.2.i.bg.1729.4 8
36.23 even 6 2592.2.i.bg.865.4 8
36.31 odd 6 2592.2.i.bh.865.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.v.1.2 ✓ 4 3.2 odd 2
2592.2.a.v.1.3 yes 4 4.3 odd 2
2592.2.a.w.1.1 yes 4 12.11 even 2 inner
2592.2.a.w.1.4 yes 4 1.1 even 1 trivial
2592.2.i.bg.865.1 8 9.4 even 3
2592.2.i.bg.865.4 8 36.23 even 6
2592.2.i.bg.1729.1 8 9.7 even 3
2592.2.i.bg.1729.4 8 36.11 even 6
2592.2.i.bh.865.2 8 36.31 odd 6
2592.2.i.bh.865.3 8 9.5 odd 6
2592.2.i.bh.1729.2 8 36.7 odd 6
2592.2.i.bh.1729.3 8 9.2 odd 6
5184.2.a.cd.1.2 4 8.5 even 2
5184.2.a.cd.1.3 4 24.11 even 2
5184.2.a.ce.1.1 4 8.3 odd 2
5184.2.a.ce.1.4 4 24.5 odd 2